# Properties

 Label 1800.4.a.f Level $1800$ Weight $4$ Character orbit 1800.a Self dual yes Analytic conductor $106.203$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$106.203438010$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 20 q^{7}+O(q^{10})$$ q - 20 * q^7 $$q - 20 q^{7} + 56 q^{11} + 86 q^{13} - 106 q^{17} + 4 q^{19} + 136 q^{23} + 206 q^{29} - 152 q^{31} - 282 q^{37} + 246 q^{41} - 412 q^{43} + 40 q^{47} + 57 q^{49} - 126 q^{53} - 56 q^{59} - 2 q^{61} + 388 q^{67} + 672 q^{71} - 1170 q^{73} - 1120 q^{77} + 408 q^{79} + 668 q^{83} - 66 q^{89} - 1720 q^{91} + 926 q^{97}+O(q^{100})$$ q - 20 * q^7 + 56 * q^11 + 86 * q^13 - 106 * q^17 + 4 * q^19 + 136 * q^23 + 206 * q^29 - 152 * q^31 - 282 * q^37 + 246 * q^41 - 412 * q^43 + 40 * q^47 + 57 * q^49 - 126 * q^53 - 56 * q^59 - 2 * q^61 + 388 * q^67 + 672 * q^71 - 1170 * q^73 - 1120 * q^77 + 408 * q^79 + 668 * q^83 - 66 * q^89 - 1720 * q^91 + 926 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 −20.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.4.a.f 1
3.b odd 2 1 600.4.a.i 1
5.b even 2 1 360.4.a.n 1
5.c odd 4 2 1800.4.f.v 2
12.b even 2 1 1200.4.a.p 1
15.d odd 2 1 120.4.a.b 1
15.e even 4 2 600.4.f.a 2
20.d odd 2 1 720.4.a.q 1
60.h even 2 1 240.4.a.g 1
60.l odd 4 2 1200.4.f.t 2
120.i odd 2 1 960.4.a.bj 1
120.m even 2 1 960.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 15.d odd 2 1
240.4.a.g 1 60.h even 2 1
360.4.a.n 1 5.b even 2 1
600.4.a.i 1 3.b odd 2 1
600.4.f.a 2 15.e even 4 2
720.4.a.q 1 20.d odd 2 1
960.4.a.k 1 120.m even 2 1
960.4.a.bj 1 120.i odd 2 1
1200.4.a.p 1 12.b even 2 1
1200.4.f.t 2 60.l odd 4 2
1800.4.a.f 1 1.a even 1 1 trivial
1800.4.f.v 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} + 20$$ T7 + 20 $$T_{11} - 56$$ T11 - 56 $$T_{17} + 106$$ T17 + 106

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T - 56$$
$13$ $$T - 86$$
$17$ $$T + 106$$
$19$ $$T - 4$$
$23$ $$T - 136$$
$29$ $$T - 206$$
$31$ $$T + 152$$
$37$ $$T + 282$$
$41$ $$T - 246$$
$43$ $$T + 412$$
$47$ $$T - 40$$
$53$ $$T + 126$$
$59$ $$T + 56$$
$61$ $$T + 2$$
$67$ $$T - 388$$
$71$ $$T - 672$$
$73$ $$T + 1170$$
$79$ $$T - 408$$
$83$ $$T - 668$$
$89$ $$T + 66$$
$97$ $$T - 926$$
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